Improve lower bound for C_3c

[sebastian-griego]

Summary

This updates the lower bound for the 4-slope Kakeya-type sum-difference constant from

[
C_{3c} \geq 1.67471
]

to

[
C_{3c} \geq 1.67473389.
]

The bound comes from a 26-point entropy construction. The rounded weights below sum to

[
0.99999999999852250171,
]

and after normalizing them, one obtains

[
\begin{aligned}
H(X) &= 1.6377686609033206,\
H(Y) &= 1.6377686609051913,\
H(X+Y) &= 1.6377686609063419,\
H(X+2Y) &= 1.6377686607925004,\
H(X-Y) &= 2.7428266885756738.
\end{aligned}
]

Thus

[
\frac{H(X-Y)}{\max(H(X),H(Y),H(X+Y),H(X+2Y))}
= 1.6747338949921000\ldots,
]

which certifies the stated lower bound.

This also fixes the apparent typo in the entropy formulation on the 3c.md page, changing C_{3b} to C_{3c}.

Certificate

X	Y	weight
1	7	1.30506564e-12
3	5	9.25745153e-7
3	6	1.52498825e-6
3	7	6.57289640e-8
5	4	0.001244422280
5	5	0.006113255152
5	6	0.000037354259
7	2	0.016223522296
7	3	0.169721258229
7	4	0.008420798270
7	5	0.011728626004
9	1	0.248375293086
9	2	0.046862624648
9	3	0.243496670072
11	0	0.010100758400
11	1	0.188770387301
11	2	0.018228788494
13	-1	0.006653795672
13	0	0.002961930500
13	1	0.020758390703
15	-2	0.000003025952
15	-1	0.000275897370
15	0	0.000020680540
17	-3	6.99564607e-12
17	-2	1.51075546e-9
17	-1	2.78909933e-9

Verification

A short Python script using decimal.Decimal with 100 digits of precision checks the certificate. It normalizes the rounded weights before computing the entropies.

#!/usr/bin/env python3
from collections import defaultdict
from decimal import Decimal, getcontext

getcontext().prec = 100
LN2 = Decimal(2).ln()

WEIGHTS = [
    (1, 7, "1.30506564e-12"),
    (3, 5, "9.25745153e-7"),
    (3, 6, "1.52498825e-6"),
    (3, 7, "6.57289640e-8"),
    (5, 4, "0.001244422280"),
    (5, 5, "0.006113255152"),
    (5, 6, "0.000037354259"),
    (7, 2, "0.016223522296"),
    (7, 3, "0.169721258229"),
    (7, 4, "0.008420798270"),
    (7, 5, "0.011728626004"),
    (9, 1, "0.248375293086"),
    (9, 2, "0.046862624648"),
    (9, 3, "0.243496670072"),
    (11, 0, "0.010100758400"),
    (11, 1, "0.188770387301"),
    (11, 2, "0.018228788494"),
    (13, -1, "0.006653795672"),
    (13, 0, "0.002961930500"),
    (13, 1, "0.020758390703"),
    (15, -2, "0.000003025952"),
    (15, -1, "0.000275897370"),
    (15, 0, "0.000020680540"),
    (17, -3, "6.99564607e-12"),
    (17, -2, "1.51075546e-9"),
    (17, -1, "2.78909933e-9"),
]


def entropy(masses):
    return sum(-p * (p.ln() / LN2) for p in masses if p)


def marginal_entropy(points, transform):
    masses = defaultdict(Decimal)
    for x, y, p in points:
        masses[transform(x, y)] += p
    return entropy(masses.values())


total = sum(Decimal(w) for _, _, w in WEIGHTS)
points = [(x, y, Decimal(w) / total) for x, y, w in WEIGHTS]

values = {
    "H(X)": marginal_entropy(points, lambda x, y: x),
    "H(Y)": marginal_entropy(points, lambda x, y: y),
    "H(X+Y)": marginal_entropy(points, lambda x, y: x + y),
    "H(X+2Y)": marginal_entropy(points, lambda x, y: x + 2 * y),
    "H(X-Y)": marginal_entropy(points, lambda x, y: x - y),
}
denominator = max(values[name] for name in ["H(X)", "H(Y)", "H(X+Y)", "H(X+2Y)"])
ratio = values["H(X-Y)"] / denominator

print(f"sum of rounded input weights = {total}")
for name in ["H(X)", "H(Y)", "H(X+Y)", "H(X+2Y)", "H(X-Y)"]:
    print(f"{name:8s} = {values[name]}")
print(f"denominator = {denominator}")
print(f"ratio       = {ratio}")

assert ratio > Decimal("1.67473389")

Expected output includes:

sum of rounded input weights = 0.99999999999852250171
H(X)     = 1.637768660903320612195658782...
H(Y)     = 1.637768660905191258025475941...
H(X+Y)   = 1.637768660906341851866383146...
H(X+2Y)  = 1.637768660792500439978929319...
H(X-Y)   = 2.742826688575673846487425741...
ratio    = 1.674733894992100060528014329...

AI assistance disclosure

The construction, patch, PR text, and verification script were generated with ChatGPT 5.5 Pro. I ran the verification script locally before submission.